ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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A two-step function shows the $L^{p}$ norm converging to the essential supremum

Example

On $[0, 1]$ with Lebesgue measure, let

$f : = 2 χ_{[0, 1 / 2]} + χ_{(1 / 2, 1]} .$

Then

$∥ f ∥_{p} = {(2^{p - 1} + \frac{1}{2})}^{1 / p},$

and therefore $∥ f ∥_{p} \to 2 = ∥ f ∥_{\infty}$ as $p \to \infty$ .

Facts & Assumptions

Given: The two-step function $f$ above.

[L1]

The $L^{p}$ norms of essentially bounded $L^{r}$ functions converge to the essential supremum ( $L^{p}$ norms converge to the essential supremum for essentially bounded $L^{r}$ functions).

Verification

Proof technique: Compute the $p$ -norms explicitly for a two-step simple function with two distinct values and let $p$ tend to infinity.

1.1

Direct computation gives [given, algebra] $∥ f ∥_{p}^{p} = \int_{0}^{1} ∣ f ∣^{p} d λ = \frac{2^{p}}{2} + \frac{1^{p}}{2} = 2^{p - 1} + \frac{1}{2} .$

2.1

Hence [L1, step 1.1, algebra] $∥ f ∥_{p} = {(2^{p - 1} + \frac{1}{2})}^{1 / p} = 2 {(\frac{1}{2} + 2^{- p - 1})}^{1 / p} .$ For $p \geq 1$ , its bracket lies between $1 / 2$ and $1$ , so its $1 / p$ power lies between $2^{- 1 / p}$ and $1$ and therefore tends to $1$ . Thus $∥ f ∥_{p} \to 2$ . This agrees with [L1] because the essential supremum of $f$ is $2$ . ∎

Depends on

$L^p$ norms converge to the essential supremum for essentially bounded $L^r$ functions

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Richard L. Wheeden and Antoni Zygmund, Measure and Integral, Theorem 8.1 (standard reference, not scraped)

A two-step function shows the LpLp norm converging to the essential supremum